International Journal of Mathematical Engineering and ... - ijmes

International Journal of Mathematical Engineering and Science 

ISSN:2277-6982 Volume 1 Issue 5 (May 2012) 

http://www.ijmes.com 

https://sites.google.com/site/ijmesjournal/ 

A NOTE ON TRIGONOMETRIC FUNCTIONS AND 

INTEGRATION 

Mehsin Jabel Atteya, Dalal Ibraheem Rasen 

Department of Mathematics 

Al- Mustansiriyah University, College of Education 

Baghdad ,Iraq. 

mehsinatteya@yahoo.com, 

dalalresan@yahoo.com 

Abstract: The main purpose of this paper is to use the idea of finding the unlimited 

integration of the product of powers of the (sinz) and (cosz)-functions, and the product of 

powers of the (tanz) and (secz)-functions to derive tow trigonometrically identities. 

Key words: Complex functions, integration, complex trigonometric functions. 

ASM Classification Number: Primary 26D05. 

§1.INTRODUCTION 

This paper is inspired and motivation by the work of [1].The theory of functions of 

complex variable , also called for brevity complex variables or complex analysis , is one of 

the most beautiful as well as useful branches of mathematics. Although originating in an 

atmosphere of mystery, suspicion and distrust ,as evidenced by the terms "imaginary" and 

"complex" present in the literature, it was finally placed on a sound foundation in the 19 th 

century through the efforts of Cauchy, Riemann, Weierstrass, Gauss and other great 

mathematicians. Today the subject is recognized as an essential part of the mathematical 

background of engineers, Physicists, mathematicians and other scientists. From the 

theoretical view –point this is because many mathematical concepts become clarified and 

unified when examined in the light of complex variable theory. From the applied viewpoint 

the theory is of tremendous value in the solution of problems of heat flow, potential theory , 

fluid mechanics, electromagnetic theory, aerodynamics, elasticity and many other field of 

science and engineering. Many authors doing in complex variables and gave some results 

about that [see (2),(3),(4)]. A. D. Sinder [5] proved that , if the complex –valued function f(t) 

is continuous on [a,b] and F′(t)=f(t) for all t in [a,b],then 

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Murray R.Spiege [6]proved that, the integration of F(z)G(z) 

is equal to F(z)G(z)- 

Where F(z) and G(z) are complex variable functions . Ruel V. Churchill , James W. Brown 

and Roger F. Verhey [7] proved that 

Whenever the path of integrations is a contour. 

From our we know study the integration of the trigonometric function (sinz cosz dz). 

The solution of above integration we can find by substituting y=sin z to obtain that. 

+c 1 

If we use the substituting y=cos z , we obtain +c 2 

It is clear from that the difference between these tow answers is constant number this implies 

to . 

sin z+cos z=c , 

where c is constant number and by substituting z=0 in the above equation we get. 

sin z+cos z=1. 

In our paper we shall is to use the idea of finding the unlimited integration of the product of 

powers of the (sin z) and (cos z)-functions, and the product of powers of the (tan z) and (sec 

z)-functions. 

§2.THE MAIN RESULTS: 

Theorem 2.1:For all complex values of z , the following 

= 

Will be true where n,m are positive integers. 

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Proof: We consider the following integration 

z cos 

z dz 

By writing this integration by the following 

z cos 

z cosz dz 

= z (1-sin 

By using the substituting y=sin z, we get 

And by using Binomial formula we get 

y 

dy 

= dy 

= dy 

= +c 1 

= c 1 

By same method we can find z cos z dz 

By writing this integration by the following 

cos 

z sin z dz 

Using the substituting y=cos z, for obtain 

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And by using Binomial formula we get 

- y dy 

= dy 

= dy 

= +c 2 

= c 2 

But the difference between tow answers is constant number, this implies to 

+ =c(n,m) 

And by choose z=0 ,we get 

C(n, m)= 

We assume that x=m+1,which implies 

= 

Thus c(n, m)= 

By same method in the proof of Theorem(2.1)we can prove the following theorem 

Theorem2.2: For all complex values of z , the following 

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, will be true where n, m are positive integers 

Proof: We consider the following integration 

We take y=tan z and y=sec z, so it easy to complete our prove by same method in 

Theorem(2.1). 

We closed our paper by the following corollaries after depended on some fundamental 

relations of exponential and trigonometric functions. 

Corollary 2.3: For all complex values of z, the following 

= 

Will be true where n, m are positive integers 

Corollary 2.4: For all complex values of z, the following 

= 

Will be true where n , m are positive integers 

Acknowledgments. The authors would like to thank the referee for her/his useful comments 

. 


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REFERENCES: 

[1] I. Dubait , Integration and Trigonometry , Dirasat , Pure Sciences , Vol .4 (1997), No .1, 

665-666. 

[2] Olariu , Silviu(2002) , Complex Numbers in N Dimensions, Hyperbolic Complex 

Numbers in Two Dimensions, North-Holland Mathematics Studies #190, Elsevier ISBN 0- 

444-51123-7, ,Pages 1-16. 

[3] Hazewinkle , M., Double and dual numbers, Encyclopedia of Mathematics 

Soviet/AMS/Kluwer,Dordrect,(1994). 

[4] Irene Sabadini (2009), Michael Shapiro & Frank Sommen , editors , Hypercomplex 

Analysis and Applications Birkhauser ISBN 9783764398927. 

[5]A.D.Sinder(1976) , Fundamentals of complex analysis for mathematics Science and 

Engineering, Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 

[6] Murray R. Spiegel(1972) ,Theory and Problems of Complex Variables, Schaum′s 

Outline Series, McGraw-Hill Book Company. 

[7] Ruel V.Churchill, James W.Brown and Roger F. Verhey(1974), Complex Variables and 

applications, 3th Edition. International Student Edition ,McGraw-Hill Kogakusha,LTD. 

. 

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